# Random reals and strongly meager sets

Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this 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 $X\subseteq \mathbb{R}$ is strong measure zero if and only if for any meager set $Y$ there exists $t\in \mathbb{R}$ such that $(t+X)\cap Y=\varnothing$.

Random real forcing is dual to Cohen forcing in the sense of measure and category. Therefore it makes sense to ask, whether:

The set of reals from generic model $\mathbb{R}\cap V$ is strongly meager after adding a single random real?

I have heard that the answer is affirmative, but I have not been able to find any published proof. Note that $\mathbb{R}\cap V$ is meager after adding a random real (see this question).

-