Extending BMO functions in parabolic setting

We prove that one can extend any $BM{O}^x$ function $a$ given in a cube in ${\mathbb{R}}^{d+1}$ to become a $BM{O}^x$ functions $\widehat{a}$ in ${\mathbb{R}}^{d+1}$ almost preserving its $[a{]}^{\mathrm{♯}}$ seminorm, which is, loosely speaking, $L_{\mathrm{\infty }}$-norm of the maximal function in $t$ and $BMO$-norm in $x$.