SPSecurity.RunWithElevatedPrivileges Not Working for Read Only Permissions Users

I have the following code in a method that generates tabbed web parts on any page in SharePoint 2010.

The problem is that it will not work for users who have Read access only on a SharePoint site.  It will work when those users have Contribute access.  

So even though I have elevated permissions in the code it does not actually elevate the permissions at the point where it is needed.

if (panel != null)
                {
                    try
                    {
                        using (SPLimitedWebPartManager wpManager = SPContext.Current.Web.GetLimitedWebPartManager(HttpContext.Current.Request.Url.ToString(), PersonalizationScope.Shared))
                        {
                            try
                            {
                                // Elevated previleges required for EXPORT and IMPORT. Else Users with normal read access will get errors.
                                SPSecurity.RunWithElevatedPrivileges(delegate()
                                {
                                    // Retrieve the web part titles in the ConfigureTabs XML string for this tab.
                                    var webPartTitles = from t in xDocument.Descendants("webPart")
                                                        where (string)t.Parent.Attribute("name") == (string)e.Item.DataItem
                                                        select (string)t.Attribute("title");

                                    foreach (string wpTitle in webPartTitles)
                                    {
                                        foreach (System.Web.UI.WebControls.WebParts.WebPart webPart in wpManager.WebParts)
                                        {
                                            // Find the matched closed web part in WebParts collection
                                            if (webPart.Title == wpTitle && webPart.IsClosed == true)
                                            {
                                                string errorMessage;
                                                //ADD EXPORT PROPERTY
                                                webPart.ExportMode = WebPartExportMode.All;
                                                MemoryStream stream = new MemoryStream();
                                                XmlTextWriter writer = new XmlTextWriter(stream, System.Text.Encoding.UTF8);

                                                // Export the closed webpart to a memory stream.
                                                wpManager.ExportWebPart(webPart, writer);

                                                writer.Flush();
                                                stream.Position = 0;

                                                XmlTextReader reader = new XmlTextReader(stream);

                                                // Import the exported webpart.
                                                System.Web.UI.WebControls.WebParts.WebPart newWebPart = wpManager.ImportWebPart(reader, out errorMessage);

                                                reader.Close();
                                                writer.Close();

                                                // Show the imported webpart.
                                                panel.Controls.Add(newWebPart);
                                                break;
                                            }
                                        }
                                    }
                                });
                            }
                            catch (Exception ex)
                            {
                                // For debugging use only.
                                Label label = new Label();
                                label.Text = "Please check your XML configuration for error. " + Environment.NewLine + ex.Message;
                                panel.Controls.Add(label);
                            }

                        }

                    }
                    catch (Exception ex)
                    {
                        // For debugging use only.
                        Label label = new Label();
                        label.Text = "Please Check SPContext.Current.Web is not null. " + Environment.NewLine + ex.Message;
                        panel.Controls.Add(label);
                    }
                }

This snippet of code was originally pulled from a microsoft technet article on creating Tabbed web parts "the correct way" but it doesn't work in all scenarios.

Is there a way to get this code working for Read/Visitors to a SharePoint site?