The diophantine exponent of the $mathbb{Z}/qmathbb{Z}$ points of $S^{d-2}subset S^d$

Assume a polynomial-time algorithm for factoring integers, Conjecture~ref{conj}, $dgeq 3,$ and $q$ and $p$ are prime numbers, where $pleq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $log(q)$ that lifts every $mathbb{Z}/qmathbb{Z}$ point of $S^{d-2}subset S^{d}$ to a $mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $mathbb{Z}/qmathbb{Z}$ points of $S^{d-2}subset S^d$.