Monomial congruence counting conjecture

Let $a,b,c$ be non-zero integers with $a,b,c$ setwise coprime, and let $d\geqslant 1$ be another integer. For a prime $q\geqslant 3$, define $N_{a,b,c,d}(L,M,N;q)$ to count triples $(l,m,n)\sim L\times M\times N$ satisfying

$$(l^a m^b n^c)^d\equiv 1\pmod q,$$

but with $l^a m^b n^c\ne 1$. Monomial congruence counting conjecture. There exists $\eta_0>0$, depending on $a,b,c,d$ but not on $q$, such that for every $L,M,N\geqslant 1$ and every $\varepsilon>0$,

$$\frac{N_{a,b,c,d}(L,M,N;q)}{\sqrt{LMN}}\ll_{\varepsilon,a,b,c,d}q^\varepsilon\left(\frac{\sqrt{LMN}}{q}+(LMN)^{-\eta_0}\right).$$

This estimate is the conjectural input needed to control error terms in the cubic moment analysis and would establish the expected asymptotic lower-bound equality in the general case. The supplied text does not state whether the conjecture has been proved or refuted.