The actually bidding process requires a form submission, so it does require a page refresh to place a bid.
I recall that you have disabled the bid confirm page as per this ticket.
Be aware that modifications to the product may affect the behavior, and I'm probably not going to be able to reproduce your issue on my test site.
So to debug this one, I think you may need to firstly determine whether the mybid function in controller.php is actually getting called under these circumstances, which you can probably do with an echo statement, maybe something like:
echo "<br>DEBUG: in the mybid function";
Once you've confirmed that this function is actually called, you'd then want to take a look at all the request variables and examine their values to see if the correct data is being passed. Maybe something like:
echo "<br>DEBUG: bid=".$bid;
My guess is that either the mybid function is not getting called, or one of these variables is still picking up the value from the previous bid, in which case you may need to reset that variable at the end of the mybid function.
I hope this helps.
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