  In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two
spuercongruences on truncated hypergeometric series and further proposed two
related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066]
extended He's formulas and confirmed the second conjecture. However, the first
conjecture is still open up to now. With the help of the creative microscoping
method and the Chinese remainder theorem for coprime polynomials, we derive
several $q$-supercongruences modulo the fourth and fifth powers of a cyclotomic
polynomial from Gasper and Rahman's summation formula for basic hypergeometric
series. As conclusions, He's first conjecture is confirmed and a more general
form of He's second conjecture is proved.
