a compact set with nonempty convex sections

Question

Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space. For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$. Given a set $Y \subseteq X$ and a vector $x_{-i} \in [0,1]^{d-1}$ denote by $Y_{x_{-i}} := \{(x_i,x_{-i}) \in Y \colon x_i \in [0,1]\}$ the $x_{-i}$-section of $Y$. Let $Y \subseteq X$ be a compact set that satisfies the following condition: for every $x \in X$ and every coordinate $i=1,2,\ldots,d$, the $x_{-i}$-section $Y_{x_{-i}}$ is nonempty and convex. Is it true that the set $Y$ is contractible?