Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$.

For $0\le p\le d$, denote by $C^p(X,i)$ the Chow $k$-variety of effective codimension $p$ algebraic cycles on $X$.

Fix affine codimension $n$ $k$-subvarieties $\mathscr{Y}:=\{Y_1,\ldots, Y_t\}$ for some fixed integers $n,t$, with $0\le n\le d$, and for an integer $0\le s\le d$ denote by $$W^p_{\mathscr{Y},s}(X,i)$$ the locus of $C^p(X,i)$ of those effective codimension $p$ $k$-cycles on $X$ each of whose components intersect each one the $Y_i$'s in $\mathscr{Y}$ in codimension at least $s$.

Is $W^p_{\mathscr{Y},s}(X,i)\subset C^p(X,i)$ an open subvariety? Is it closed?