Bayesian interpretation of Square-Root LASSO

There are various sources that discuss the parallel between the LASSO and their Bayesian interpretation, e.g., this SE question and various other, showing, that the $$\ell_1$$ regularization corresponds with an i.i.d. Laplace prior for the $$\beta$$, while $$y$$ is assumed as $$y \sim \mathcal{N}(X\beta, \sigma I)$$. From here, the LASSO follows from minimizing the log-likelihood of $$y$$.

My question: can we follow similar arguments to devise Bayesian interpretation for the Square-Root LASSO? (Which is defined as

$$\hat\beta_{SL} = \arg\min_{\beta} \{ \hat Q(\beta)\}^{1/2} + n^{-1}\lambda ||\beta||_{1}$$

here $$\hat Q(\beta) = n^{-1}\sum_{i=1}^n (y - x_i'\beta)^2$$, (Belloni et al.)).

What would the distribution of $$y$$ look like, if we followed the same log-likelihood arguments?

The focus on the question is on the form of the residual sum of squares term $$\hat Q$$ and modifying it with a square-root in the log-likelihood form, not about the penalty term. Hence, this is not a duplicate. For me, it's not trivial that the square-root term would carry over to the gaussian density straightforwardly and whether it would still remain a proper density function, and what would be the caveats of that form.