Random reals and strongly meager sets

It is well known (but unpublished as far as I know), that adding a single Cohen real makes the set of reals from the ground model strong measure zero, see this Mathoverflow question.

The notion of strong measure zero sets has its dual concept in the category branch — strongly meager sets. A set X\subseteq\mathbb{R} is strongly meager if for any null set Y there exists t\in\mathbb{R} such that (t+X)\cap Y=\varnothing. One can see duality of these notions due to Galvin-Mycielski-Solovay Theorem which states that a set X is strong measure zero if and only if for any meager set Y there exists t such that (t+X)\cap Y=\varnothing.

Therefore I asked whether the set of reals from the ground model \mathbb{R}\cap V is strongly meager after adding a single random real.

The answer is affirmative and I have heard it was known, but unpublished. Finally my advisor Prof. Tomasz Weiss came up with the following proof.

See also my poster on this topic (7th Young Set Theory workshop, Będlewo 2014).