Split $g_k$ into $g_k^-+g_k^+$ where $g_k^-$ is the part lying to the left from the median of $X_k$ (it depends on $k$ nownow; note also that $X_k$ is not a misprint: we, indeed, split the shot according to the median of the step) and $g_k^+$ is the part lying to the right of it. Split $f_k$ in the same way. It is easy to see that $g_k^+$ is dominated by $f_k^+$.
Now we shall run the induction on the walk length. Suppose that the bound for all shorter walks is already established and we want to investigate some walk of the currently suspicious length corresponding to some $n$. Chopping off the $g^-$ part, we see that our main task is to bound
$$
Q=g_1^++f_1*g_2^++f_1*f_2*g_3^++f_1*f_2*f_3*g_4^++f_1*f_2*f_3*f_4*g_5^++f_1*f_2*f_3*f_4*f_5*g_6^++\dots
$$
Write
$$
\frac 32 Q=
\\
(\frac {g_1^+}2+f_1*g_2^+)+(f_1*f_2)*(\frac{g_3}2+f_3*g_4^+)+(f_1*f_2)*(f_3*f_4)*(\frac{g_5}2+f_5*g_6^+)+\dots
\\
+f_1*[(\frac {g_2^+}2+f_2*g_3^+)+(f_2*f_3)*(\frac{g_4}2+f_4*g_5^+)+(f_2*f_3)*(f_4*f_5)*(\frac{g_6}2+f_6*g_7^+)+\dots]
\\
+\text{ a few ($\le 8$, say) individual endpoint terms due to the boundary effect and possibly wrong parity}
$$$$
\frac 32 Q=
\\
(\frac {g_1^+}2+f_1*g_2^+)+(f_1*f_2)*(\frac{g_3^+}2+f_3*g_4^+)+(f_1*f_2)*(f_3*f_4)*(\frac{g_5^+}2+f_5*g_6^+)+\dots
\\
+f_1*[(\frac {g_2^+}2+f_2*g_3^+)+(f_2*f_3)*(\frac{g_4^+}2+f_4*g_5^+)+(f_2*f_3)*(f_4*f_5)*(\frac{g_6^+}2+f_6*g_7^+)+\dots]
\\
+\text{ a few ($\le 8$, say) individual endpoint terms due to the boundary effect and possibly wrong parity}
$$
Estimate the individual terms by $Gn^{-1/2}$ each and note that each of the long lines represents a shooting walk of smaller length with steps $f_k*f_{k+1}\in \mathcal F_{2n}$ and the corresponding shots $\frac{g_k^+}2+f_k*g_{k+1}^+$ (the initial convolution with $f_1$ in the second long line is harmless for bounding the maximum and can be safely forgotten).
