Before revision:  The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-monotonic language as well as  the injectivity problem which is undecidable for PFA over letter-monotonic languages.
After revision:  The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-monotonic language as well as the injectivity problem which is undecidable for PFA over letter-bounded languages.