Permutation conjecture for prime-exponent LCMs of distinct covering systems

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m\geq 2$, let $a$, $b$, and $c$ be positive integers, and let $L$ denote the least common multiple of its moduli. For a permutation $(a_1,b_1,c_1)$ of $(a,b,c)$ with $a_1\geq b_1\geq c_1$, the permutation conjecture. If there exists a distinct covering system with least modulus $m$ and

$$L=2^a3^b5^c,$$

then there exists a distinct covering system with least modulus $m$ and

$$L=2^{a_1}3^{b_1}5^{c_1}.$$

The conjecture motivates restricting the analysis to exponents ordered as $a\geq b\geq c$. The supplied text gives no resolution status.