Characterization of mutual singularity of Sobolev spaces on Laakso-type fractals

Let $p_1,p_2\in(1,\infty)$, let $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ be the corresponding Sobolev spaces, and let $U_{p_1,+}$ and $U_{p_2,+}$ be their optimal potential functions. Mutual singularity conjecture. The Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ are mutually singular if and only if $U_{p_1,+}\neq U_{p_2,+}$. The preceding discussion establishes that unequal optimal potentials imply non-membership of the associated distinguished function in the other Sobolev space, while the converse is proposed as a complete characterization of mutual singularity.