The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. Math. 39, 1952.

Observe that if $m: \mathcal{P}(X) \to [0, 1]$ is a diffused (vanishes on singletons) probability measure, then $F_m: 2^{X} \to [0, 1]$ defined by $F_m(1_A) = m(A)$ is a sequentially continuous (whenever $A_n \to A$, $m(A_n) \to m(A)$) but discontinuous map (the preimage of $\{0\}$ is not closed in $2^X$).

Question: Let $\kappa$ be the least cardinal such that there is a sequentially continuous but discontinuous map from $2^{\kappa}$ to $[0, 1]$. Must $\kappa$ admit a total diffused probability measure? In modern terms, is $\kappa$ real valued measurable?

The only partial result that I know here appears in D. Choodnovsky: Sequentially continuous mappings of product spaces, Seminaire D'Analyse Fonctionnelle Ecole Polytechnique, 1977-78, Exp. no. 4, pp 1-15 where, among other things, it is shown that such a cardinal must admit an $\aleph_1$-saturated sigma ideal.

The above paper is pretty old and I couldn't find any recent survey on this problem so I am wondering about the current status of the problem. Thanks!