The template conjecture for monochromatic block sets

For a positive integer $m$, write $[m]=\{1,2,\ldots,m\}$. A template over $[m]$ is a non-decreasing word $\tau\in[m]^\ell$ for some positive integer $\ell$. A block set with template $\tau$ in $[m]^n$ is formed from pairwise disjoint blocks $I_1,\ldots,I_\ell\subset[n]$, fixed coordinates outside their union, and all rearrangements of $\tau$ assigned to the blocks. Its degree is $\sum_{j=1}^\ell|I_j|$.

Template conjecture. For every pair of positive integers $m$ and $k$, and every template $\tau$ over $[m]$, there exist positive integers $n$ and $d$ such that every $k$-colouring of $[m]^n$ contains a monochromatic block set of degree $d$ with template $\tau$.

This conjecture is equivalent to the block-permutation conjecture: the template $12\ldots m$ gives the latter, while the general case follows by applying it to a colouring induced through the template. The paper records that the conjecture is true for $m=1$, and for all templates when $m=2$, but remains open in general.