True/false: higher forward gain reduces disturbance

True, provided the gain is increased in the forward path before the point where the disturbance enters (and stability is kept).

Let the disturbance $D(s)$ enter between $G_1(s)$ and $G_2(s)$, with feedback $H(s)$. With $R = 0$:

$$C_D(s) = \frac{G_2}{1+G_1G_2H}\,D(s) \approx \frac{D(s)}{G_1H} \quad (|G_1G_2H| \gg 1)$$

Increasing the forward gain $G_1$ increases the denominator, so the output caused by the disturbance falls. Meanwhile the output due to the reference, $C_R = \dfrac{G_1G_2}{1+G_1G_2H}R \approx R/H$, is almost unchanged.

Example: raising $G_1G_2H$ from 9 to 99 cuts the disturbance effect from $1/10$ to $1/100$ of the open-loop value.

Note: increasing only $G_2$ (after the disturbance) does not help, since $C_D \to D/(G_1H)$; and very high gain may reduce damping or cause instability.