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 follows from Dubins–Schwarz's theorem that there exists some Brownian motion $B$ (defined on some suitable probability space) s.t. $X$ is a time-change of $B$. More precisely, there exists 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

$$\quad\Law(Y)=\Law\bigl((B_{G_t})_{t\ge 0}\bigr), \quad\mbox{with}\quad  G_t:=\int_0^t g_s^2 \,ds?$$