For any Borel sets $A$ and $B$ such that $p_A:=P(X\in A)\in(0,1)$ and $r_B:=P(Y\in B)\in(0,1)$, let $q_A:=1-p_A$, $s_B:=1-r_B$, \begin{equation} f(X):=\pm\frac{1_{X\in A}-p_A}{\sqrt{p_A q_A}}, \quad g(Y):=\frac{1_{Y\in B}-r_B}{\sqrt{r_B s_B}}. \end{equation} Then the condition $\rho_m(X;Y) < 1/d$ implies \begin{equation} |P(X\in A,Y\in B)-P(X\in A)P(Y\in B)|\le\tfrac1d\,\sqrt{P(X\in A)P(X\notin A)P(Y\in B)P(Y\notin B)}. \end{equation} The latter inequality holds even when $p_A\in\{0,1\}$ and/or $r_B\in\{0,1\}$.

For any Borel sets $A$ and $B$ such that $p_A:=P(X\in A)\in(0,1)$ and $r_B:=P(Y\in B)\in(0,1)$, let $q_A:=1-p_A$, $s_B:=1-r_B$, \begin{equation} f(X):=\pm\frac{1_{X\in A}-p_A}{\sqrt{p_A q_A}}, \quad g(Y):=\frac{1_{Y\in B}-r_B}{\sqrt{r_B s_B}}. \end{equation} Then the condition $\rho_m(X;Y) < 1/d$ implies \begin{equation} |P(X\in A,Y\in B)-P(X\in A)P(Y\in B)|\le\tfrac1d\,\sqrt{P(X\in A)P(X\notin A)P(Y\in B)P(Y\notin B)}. \end{equation} The latter inequality holds even when $p_A\in\{0,1\}$ and/or $r_B\in\{0,1\}$.

So, if $d$ is large, then the dependence between $X$ and $Y$ is very weak.