to activate the checkbox you should tap systematically 2 times (not several).
We've found an unexpected value in the checkboxes definition from the AcroForm inner part (when opened first time, even if the graphical representation is "off", the inner code says "true").
We're better analyzing the status and working on a solution or workaround.
Dear Marat, to provide you a proof of our previous answer: you should try the linked file.
I've just opened it using AcrobatPro and reset the first two checkboxes' rows.
Playing with checkboxes from that two rows you shall find the expected result.