Long-period radio transients (LPTs) are periodic radio sources showing pulsed emission on time-scales from minutes to hours. The underlying sources behind this emission are currently unclear. There are two leading candidates: neutron stars or white dwarfs. Neutron stars could emit at LPT time-scales as magnetars, binaries, or precessing sources. White dwarfs on the other hand have only been observed to emit in radio as binary systems with companions that provide charged particles through their wind. A key distinction is that an optical counterpart is much more likely in the white dwarf scenario. GPM J1839$-$10 is an LPT with a radio period of 21 min for which the white dwarf scenario has been favoured, but no optical counterpart is confirmed. Using HiPERCAM, a high-speed multicolour photometer that observes simultaneously in $u_sg_sr_si_sz_s$ filters, we probe the existence of a white dwarf in GPM J1839$-$10. We do not directly detect a white dwarf, but cannot rule out its presence given the uncertain distance and reddening of GPM J1839$-$10. On the other hand, we find evidence in our data for periodic behaviour in harmonics of the radio period, as expected from the white dwarf scenario.