The spanning-tree factorization conjecture for Sierpinski gaskets

Let $SG_d(n)$ be the $d$-dimensional Sierpinski gasket at stage $n$, and let $f_d(n)=N_{ST}(SG_d(n))$ denote its number of spanning trees. Define exponents $\alpha_d(n)$, $\beta_d(n)$, and $\gamma_d(n)$ by

$$\alpha_d(n)=\frac{d-1}{2}\bigl[(d+1)^n-1\bigr],$$ $$\beta_d(n)=\frac{d-1}{2d}\bigl[(d+1)^{n+1}+dn+d-1\bigr],$$ $$\gamma_d(n)=\frac{d-1}{2d}\bigl[(d+1)^n-dn-1\bigr].$$

Spanning-tree factorization conjecture. The number of spanning trees is

$$f_d(n)=2^{\alpha_d(n)}(d+1)^{\beta_d(n)}(d+3)^{\gamma_d(n)}.$$

This conjecture extends the simple prime-factor formulas established for the two-, three-, and four-dimensional Sierpinski gaskets. The stated exponents are positive integers for positive integer $d$ and non-negative integer $n$, and the formula agrees at $n=0$ with the spanning-tree count $(d+1)^{d-1}$ for the complete graph $SG_d(0)$. A general proof or explanatory method for the unexpectedly simple solution is not supplied.