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Higgs precision program at future lepton collider aims at (sub) percent levelprecision measurement of the Higgs properties, shedding light to new physicsthrough the Higgs lamppost. Amongst many exclusive Higgs channels that can bemeasured precisely, the $WW$-fusion to Higgs with subsequent decays into $b\barb$ final state are of particular importance. This channel provide leadingconstrains on Higgs total width in the $\kappa$-framework and greatly improvesthe constraints in the EFT framework as a distinct production mode other thanthe Higgsstrahlung process. We argue in this paper that, there are twointerference effects both affects the physical information one can extract fromthe precision measurements. One takes place at quantum level from theinterference between the two amplitudes that amounts to $-10\%$ of the$WW$-fusion signal strength, failing to take into account which will result ina $4$-$5\sigma$ discrepancy between theory and measurement. The other takesplace at the classical level from the global fitting procedure where the $ZH$process is the dominant background with its cross section around six timeslarger than the $WW$-fusion signal. Despite that $ZH$ process can be measuredto great precision at future lepton colliders, failing take this interplay inthe coupling extraction will result in a 100\% too aggressive constraints on$\kappa_W$, the Higgs coupling to $W$-boson pairs. This effect will also beimportant for lepton colliders running at slightly higher energies where thephase space overlap are still sizable between the two processes.

Double interference effects in Higgs precision tests in the $e^-e^+\to \nu\bar \nu h$ process