Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a DVR) and $\mathcal Z\subset \mathcal X$ a closed subscheme flat over $C.$ By the smooth proper base change we have $H^r(\mathcal X_{\bar{c}}, \mathbb Z_l)\stackrel{\phi}{\simeq} H^r(\mathcal X_{\bar{\eta}}, \mathbb Z_l)$ for all $c\in C,$ $r\geq 0$ and $l\neq p.$ Is it known that $$cl(\mathcal Z_{\bar{\eta}}) = \phi(cl(\mathcal Z_{\bar{c}}))$$ in $H^{2d}(\mathcal X_{\bar{\eta}},\mathbb Z_l)$ (where $cl$ is the cycle map and $d = \mathrm{dim}_k(\mathcal Z_{c})) ?$


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