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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 ofLy-$\alpha$ transmission as a precision probe of reionisation's end stages. Inthis paper, we assemble a sample of 67 quasar sightlines at $z>5.5$ with highsignal-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 onVLT/X-Shooter which obtained deep (SNR $>20$ per pixel) spectra of 30 quasarsat $z>5.7$. We carefully account for systematics in continuum reconstruction,instrumentation, and contamination by damped Ly-$\alpha$ systems. We presentimproved measurements of the mean Ly-$\alpha$ transmission over $4.9<z<6.1$.Using all known systematics in a forward modelling analysis, we find excellentagreement between the observed Ly-$\alpha$ transmission distributions and thehomogeneous-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 areruled out by excess Ly-$\alpha$ transmission scatter at $z\geq5.4$ with highconfidence ($>3.5 \sigma$). Our results indicate that reionisation-relatedfluctuations, whether in the UVB, residual neutral hydrogen fraction, and/orIGM 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 toreionisation.

Hydrogen reionisation ends by $z=5.3$: Lyman-$\alpha$ optical depth measured by the XQR-30 sample