A note on the variance of the square components of a normal multivariate within a Euclidean ball

We present arguments in favour of the inequalities $var(X_n^2|X \in B_v(\rho)) \le 2\lambda_n E[X_n^2|X \in B_v(\rho)]$, where $X \sim N_v(0,\Lambda)$ is a normal vector in $v\ge 1$ dimensions, with zero mean and covariance matrix $\Lambda = \diag(\lambda)$, and $B_v(\rho)$ is a centered $v$-dimensional Euclidean ball of square radius $\rho$. Such relations lie at the heart of an iterative algorithm, proposed in ref. [1] to perform a reconstruction of $\Lambda$ from the covariance matrix of $X$ conditioned to $B_v(\rho)$. In the regime of strong truncation, i.e. for $\rho \lesssim \lambda_n$, the above inequality is easily proved, whereas it becomes harder for $\rho \gg \lambda_n$. Here, we expand both sides in a function series controlled by powers of $\lambda_n/\rho$ and show that the coefficient functions of the series fulfill the inequality order by order if $\rho$ is sufficiently large. The intermediate region remains at present an open challenge.