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This should be possible by modifying the default Sitefinity workflow. I will verify that and will get back to you as quickly as possible.
I investigated the matter, and unfortunately our content lifecycle won't allow such customization. Let me explain why. In Sitefinity there are three lifecycle states of an item – master, temp and live. When you open a page for editing, you checkout a temporary version of the page and perform a change on it. On Publish the temp is checked in, and changes are applied to the master. After that the temporary version of the page gets deleted. On theory, Publish, as a workflow activity can be modified. At the moment when an item is published, the edit page form is closed and you get back to Pages (you can see that if you open the default workflow for editing). There's a checkbox, which specifies whether the form will be closed after publishing. If you uncheck it, you will be able to edit the page, click on publish and keep the Page Editor open, but as soon as you try to perform another operation on the page (for example add a widget), you will get a Page is locked error message. The reason for this lay in the fact that on Publish, as I already mentioned, the temp item is deleted. This temp item has a unique id, so when you try to add another widget to the page, Sitefinity practically tries to modify an object that doesn't exist, which is not possible. The page will be available for edit after the next checkout, but this time the temp version will be different and will have a different id, which can't simply replace the old temp.
Another option, which I considered, was to change the RedirectUrl, that is set to the Publish button on the Edit page. Unfortunately, this would require a custom Editor toolbar and such, cannot be set through the configurations.
As a workaround I can offer you to use the Save as draft button, which will allow you to save changes and remain of the currently edited page.
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