Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of codimension $d$, modulo rational equivalence.

I am interested in the linear subspace of $\mathbb Q\mathrm{CH}^d(X)$ which is generated by the subvarieties $Z\subseteq X$ of codimension $d$ which are locally complete intersections, so those which are locally the zero set of exactly $d$ regular functions. Let us denote this subspace by $\mathbb Q\mathrm{CH}^d_{\mathrm{lci}}(X)$. Then the question is:

Are Chow groups generated by local complete intersections? I.e. does equality $$\mathbb Q\mathrm{CH}^d_{\mathrm{lci}}(X) = \mathbb Q\mathrm{CH}^d(X)$$ hold?

If for instance $d=1$, equality holds indeed, as $X$ is smooth. I suspect this not so in general for $d\geq 2$... but where to look for a counter example?