Toda-prime lower bound for certain square-free multiples of 3

Let $T(n)$ be the set of Toda primes of $n$, where a Toda prime is an odd prime $p$ satisfying

$$p-1\mid 4n,\qquad \gcd\left(p,\frac{4n}{p-1}\right)=1.$$

Write $t(n)=|T(n)|$. Toda-prime lower-bound conjecture. Let $n$ be an odd, square-free multiple of $3$. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then

$$t(n)\geq 4.$$

This conjecture is the stated snag in the authors' attempted proof of the one-Toda-prime question. It supplies a conditional theorem-like lower bound for a structured family of integers, while the general existence question remains open.