Weaker type-2 growth conjecture for the cryptographic tensor-product space

Let $n$, $k$, and $\tilde k'$ be the parameters in the construction, let $\mathcal{S}_1^{\tilde k'\!, n}$ denote the corresponding trace-class space, and let $\ell_2^{k\tilde k'}$ be Euclidean space. For the restricted type-2 constant $\mathrm{T}_2^{(m)}$ and logarithmic comparison $\lesssim_{\log}$ used in the paper, define

$$X=\left(\mathcal{S}_1^{\tilde k'\!, n}\otimes_{(\varepsilon,\pi)_{1/2}}\mathcal{S}_1^{\tilde k'\!, n}\right)\otimes_\varepsilon\ell_2^{k\tilde k'}.$$

Weaker-form conjecture. There exists a dimension-independent constant $\beta<1$ such that

$$\mathrm{T}_2^{(n^2)}(X)\lesssim_{\log}n^\beta.$$

This weaker estimate would still imply the desired bounds for position-based cryptography. The source gives no evidence that the conjecture has been resolved.