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Here is the reply from IDM support:

We have tested it more closely.

Indeed if you not simply restart browser but close and open again, such problem persists.

But IDM add-on does not have code that deletes elements from popup menu, even for its own elements. But Column
Reader has several times deletion of some elements from menu, and it uses some complex ID of elements, so they
could easily mix something.

We are sure that this but is only from Column Reader, you should contact their support.

Ta opinia jest o wcześniejszej wersji dodatku (5.2). 

I recently encountered a similar problem in the ThunderBird mail program. The problem would be present only one in three times after start-up, but that was sufficient for fixing it. In ThunderBird, the reason for failure was that the XUL overlay code still isn't compatible to bootstrapped extensions, even though these have been around for six years now. Merely adding an event handler to the content context menu at an early stage of program start-up would prevent other items from being added to this menu.

I have never encountered this problem in Firefox, but I think it likely has the same cause when it occurs. The fix which worked in ThunderBird will be provided in the next update of Column Reader.

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When both addons are installed, many items in the Firefox's right-menu will disappear.

Ta opinia jest o wcześniejszej wersji dodatku (5.2). 

I am not going to install IDM CC, because it is loaded with dll's and accesses the Windows registry in order to determine which menu items should be visible. I just checked that there are no conflicting ID's. My guess is that the developers of IDM CC made an assumption about the menu structure which no longer holds when Column Reader is installed. My knowledge of XUL overlays is insufficient to be able to suggest a fix by merely reading the code. Perhaps the issue is gone after restarting Firefox (different installation order)?