Recently several authors (cf. [5], [6], [8], [9]) have established for arbitrary positive numbers c_1,\cdots, c_k the inequality \begin{equation*}\tag{1}P\{|X_1| \leqq c_1,\cdots, |X_k| \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i| \leqq c_i\}\end{equation*} valid for a random vector X = (X_1,\cdots, X_k) having a multivariate normal distribution with mean values 0 and with an arbitrary covariance matrix. A question then arises whether also an analogue to (1) for multivariate Student distributions holds true, i.e. the inequality \begin{equation*}\tag{2}P\{|X_1|/S_1 \leqq c_1,\cdots, |X_k|/S_k \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i|/S_i \leqq c_i\}\end{equation*} where X = (X_1,\cdots, X_k) is as before, while S_i = (\sum^p_{\nu=1} Z^2_{i\nu})^{\frac{1}{2}}, i = 1,\cdots, k, where Z_\nu = (Z_{i\nu},\cdots, Z_{k\nu}), \nu = 1,\cdots, p, is a random sample of p vectors, which are mutually independent and independent of X, and each of which has, in the simplest case, the same normal distribution as X. More generally, the Z_\nu's have some normal distributions with mean values 0 and with some covariance matrices which need not coincide with that of X and even need not be identical. A certain proof of (2) was presented by A. Scott [6] but we shall give here a counterexample showing that, unfortunately, this proof is incorrect. However, if the correlations between X_i and X_j have the form \lambda_i\lambda_j\rho_{ij} (i,j = 1,\cdots, k; i \neq j) where |\lambda_i| \leqq 1 (i = 1,\cdots, k) and where \{\rho_{ij}\} is any fixed correlation matrix, and if the correlations between Z_{i\nu} and Z_{j\nu} have the form \tau_{i\nu}\tau_{j\nu} (i,j = 1,\cdots, k; i \neq j; \nu = 1,\cdots, p) where |\tau_{i\nu}| < 1(i = 1,\cdots, k; \nu = 1,\cdots, p), we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each |\lambda_i| and each |\tau_{i\nu}|; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.