Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:=\int_0^t \alpha_s \,dW_s,\quad Y_t:=\int_0^t \beta_s\, dW_s,\quad \forall t\ge 0.$$
It is known, e.g. by Dubins–Schwarz's theorem, that there exist some Brownian motion $B$ (with respect to some suitable filtration) and a non-negative stochastic process $h$ s.t.
$$\DeclareMathOperator\Law{Law}\Law(X)=\Law\bigl((B_{H_t})_{t\ge 0} \bigr),\quad\mbox{with}\quad H_t:=\int_0^t h_s^2\,ds.$$
Define further $g_t:=\max(h_t, 1)$ for all $t\ge 0$. Can we prove
$$\quad\Law(Y)=\Law\bigl((B_{G_t})_{t\ge 0}\bigr), \quad\mbox{with}\quad G_t:=\int_0^t g_s^2 \,ds?$$
