The almost-commuting Laplacians conjecture

Let $\boldsymbol{\mathrm{L}}_1$ and $\boldsymbol{\mathrm{L}}_2$ be Laplacian matrices. Let $J(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)$ measure their commutator, let $C(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)=J(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)$, and let $C_L(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)$ denote the corresponding distance between the Laplacians and the nearest commuting Laplacians with the same structure.

Almost-commuting Laplacians conjecture. There exists a function $\delta(\epsilon)$ satisfying

$$\lim_{\epsilon\rightarrow 0}\delta(\epsilon)=0,$$

such that

$$C_L(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)\leq \delta\bigl(J(\boldsymbol{\mathrm{L}}_1,\boldsymbol{\mathrm{L}}_2)\bigr).$$

Equivalently, almost-commuting Laplacians should be close to commuting Laplacians. The paper gives empirical evidence from random Laplacian pairs, while noting that the relevant generalization of Lin's theorem remains a subject for future theoretical research; the conjecture is refuted according to the supplied status evidence.