Obtain an analytic solution for the spectral degree exponent of the path with double fork graph

Determine an analytic solution for the spectral degree exponent of the path with double fork graph $A_N$, whose value is defined by the equation $3\cdot 2^q=2+3^q$.

Background

The path with double fork graph $A_N$ has a fixed maximum degree and spectral radius for every path length parameter $N$. Consequently, its SDE is independent of $N$ and is determined by the transcendental equation $3\cdot 2^q=2+3^q$.

The paper reports only a numerical approximation, $q\approx 2.36864$, and explicitly notes that the equation has no analytically known solution. Finding an exact analytic characterization would resolve the stated gap for this graph family.