  The presence of excess scatter in the Ly-$\alpha$ forest at $z\sim 5.5$,
together with the existence of sporadic extended opaque Gunn-Peterson troughs,
has started to provide robust evidence for a late end of hydrogen reionisation.
However, low data quality and systematic uncertainties complicate the use of
Ly-$\alpha$ transmission as a precision probe of reionisation's end stages. In
this paper, we assemble a sample of 67 quasar sightlines at $z>5.5$ with high
signal-to-noise ratios of $>10$ per $\leq 15$ km s$^{-1}$ spectral pixel,
relying largely on the new XQR-30 quasar sample. XQR-30 is a large program on
VLT/X-Shooter which obtained deep (SNR $>20$ per pixel) spectra of 30 quasars
at $z>5.7$. We carefully account for systematics in continuum reconstruction,
instrumentation, and contamination by damped Ly-$\alpha$ systems. We present
improved measurements of the mean Ly-$\alpha$ transmission over $4.9<z<6.1$.
Using all known systematics in a forward modelling analysis, we find excellent
agreement between the observed Ly-$\alpha$ transmission distributions and the
homogeneous-UVB simulations Sherwood and Nyx up to $z\leq5.2$ ($<1 \sigma$),
and mild tension ($\sim 2.5 \sigma$) at $z=5.3$. Homogeneous UVB models are
ruled out by excess Ly-$\alpha$ transmission scatter at $z\geq5.4$ with high
confidence ($>3.5 \sigma$). Our results indicate that reionisation-related
fluctuations, whether in the UVB, residual neutral hydrogen fraction, and/or
IGM temperature, persist in the intergalactic medium until at least $z=5.3$
($t=1.1$ Gyr after the Big Bang). This is further evidence for a late end to
reionisation.
