Sublinear type-2 growth for the identity into the smallest valid norm

Let $X$ be a valid norm, meaning a norm on $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$ satisfying

$$\|x\|_X\gtrsim\|x\|_{X^{iii}}$$

for every $x$, together with

$$\left\|\lvert i\rangle\otimes\lvert j\rangle\otimes\left(V\lvert ij\rangle\otimes\operatorname{Id}_{\ell_2^{\tilde k'}}\right)\right\|_X\leq1$$

for every $V\in B_{M_{n^2k,\tilde k'}}$ and $i,j=1,\ldots,n$. Let $\operatorname{Id}:X\to X^{iii}$ be the identity map between the same underlying vector space equipped with these norms, and let $\mathrm{T}_2^{(n^2)}$ denote its type-2 constant restricted to $n^2$ vectors. Identity-map conjecture. There exists a valid norm $X$ and a dimension-independent constant $\beta<1$ such that

$$\mathrm{T}_2^{(n^2)}\left(\operatorname{Id}:X\to X^{iii}\right)\lesssim_{\log}n^\beta.$$

This is the final and weakest form proposed in the source; it would imply the desired lower bounds for position-based cryptography. No resolution is given.