Cooper's Lucas congruence conjecture for modular-form sequences

For $q=\exp(2\pi i\tau)$ with $\operatorname{Im}(\tau)>0$, let $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$ be the Dedekind eta function. Define the sequences $T_{11}(n)$, $T_{14,\pm}(n)$, $T_{14,\pm\epsilon}(n)$, $T_{15,\pm}(n)$, $T_{15,\pm\epsilon}(n)$, and $T_{24}(n)$ by the generating-function identities and eta-quotients in the statement below; here $i^2=-1$ and $\sqrt{32}$ denotes the positive real square root. A sequence satisfies the Lucas congruence modulo $p$ if $T(pn+j)\equiv T(n)T(j)\pmod p$ for every $n\geq 0$ and $0\leq j<p$, where $p$ is prime.

Cooper's conjecture. The sequences $\{T_{14,\pm}(n)\}$ satisfy the Lucas congruence modulo $p$ if and only if $p=2$ or $p\equiv1,7\pmod{8}$, and the sequences $\{T_{15,\pm\epsilon}(n)\}$ satisfy the Lucas congruence modulo $p$ if and only if $p=2$ or $p\equiv1\pmod{4}$.

These are experimentally proposed cases from Cooper's study of Apéry-like sequences associated with modular forms. The supplied statement also defines further sequences at levels $11$, $14$, $15$, and $24$, but does not assert Lucas-congruence claims for all of them. The status of the stated congruence conditions is not resolved in the supplied material.