  Thermal signatures of fractionalized excitations are a fingerprint of quantum
spin liquids (QSLs). In the $J_{eff}=1/2$ honeycomb magnet $\alpha$-RuCl$_3$, a
QSL state emerges upon applying an in-plane magnetic field $H_{||}$ greater
than the critical field $H_{C2} \approx$ 7 T along the a-axis, where the
thermal Hall conductivity ($k_{XY}/T$) was reported to take on the
half-quantized value $k_{HQ}/T$. This finding was discussed as a signature of
an emergent Majorana edge mode predicted for the Kitaev QSL. The $H_{||}$- and
$T$-range of the half-quantized signal and its relevance to a Majorana edge
mode are, however, still under debate. Here we present a comprehensive study of
$k_{XY}/T$ in $\alpha$-RuCl$_3$ with $H_{||}$ up to 13 T and $T$ down to 250
mK, which reveals the presence of an extended region of the phase diagram with
$k_{XY}/T \approx k_{HQ}/T$ above $H_{C2}$, in particular across a plateau-like
plane for $H_{||}$ > 10 T and $T$ < 6.5 K. From 7 T up to $\sim$10 T,
$k_{XY}/T$ is suppressed to zero upon cooling to lowest temperature without any
plateau-like behavior and exhibits correlations with complex anomalies in the
longitudinal thermal conductivity ($k_{XX}$) and magnetization around 10 T. The
results are in support of a topological state with a half-quantized $k_{XY}/T$
and suggest an interplay with crossovers or weak phase transitions beyond
$H_{C2}$ in RuCl$_3$.
