Totient-threshold non-representability conjecture for anharmonic-oscillator Stokes multipliers

Let $M$ be an integer for which the Stokes multiplier $C_M$ is defined, and let $\varphi$ denote Euler's totient function. Consider integer exponents $p$, $q_j$, $r$, $s$, and $t$ satisfying

$$|p|,|q_j|,|r|,|s|,|t|\leq 10^3.$$

Totient threshold conjecture. If $\varphi(M-1)/2\geq 2$, then $C_M$ does not admit a representation

$$C_M^{p}\cdot\prod_j\Gamma\left(\frac{a_j}{M-1}\right)^{q_j}\cdot\pi^r=2^s\cdot3^t\cdot(\text{algebraic number}).$$

The conjecture formalizes the observed failure of bounded-exponent PSLQ representations once at least two independent Gamma transcendentals are expected; its status beyond the tested finite range remains open.