Why do I get the failure «The requested package mager2pro/stripe could not be found in any version» on the «Readiness Check» step while upgrading through the web interface?

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Stripe is one of two extensions installed, and is the only one that is causing the system to halt the upgrade process. (See image)
How can this be corrected?

It is a bug in Magento 2.
You can workaround it by moving the mage2.pro repository declaration from the global Composer’s COMPOSER_HOME/config.json file to the Magento’s root composer.json file.
I have already done it for your test store (http://lexifir.www67-209-115-135.a2hosted.com):

Another solutution is to use the command line (console) interface for the Magento upgrading.