The Faber–Zagier conjecture for tautological relations on moduli spaces of curves

Let $\tau_a$ denote the cotangent-line classes used to define intersection numbers on the moduli spaces of stable curves, and let $\boldsymbol{\beth}$ denote the tautological ring generated by the classes $\boldsymbol{\beth}_j$ on ${\mathcal M}_{3\ell-1}$. Define rational numbers $a_j\in {\mathbb Q}$ by

$$\sum_{j=1}^\infty a_j t^j=-\log\left(\sum_{m=0}^\infty\frac{(6m)!}{(2m)!(3m)!}t^m\right).$$

Then the coefficient of $t^\ell$ in

$$\exp\left(\sum_{j=1}^\infty a_j\kappa_jt^j\right)\in\big({\mathbb Q}[\kappa_1,\kappa_2,\dots]\big)[[t]]$$

for each $\ell\geq 1$ gives the unique codimension-$\ell$ tautological relation among the $\kappa$-classes on ${\mathcal M}_{3\ell-1}$.