  In this paper, we study statistical inference in functional quantile
regression for scalar response and a functional covariate. Specifically, we
consider a functional linear quantile regression model where the effect of the
covariate on the quantile of the response is modeled through the inner product
between the functional covariate and an unknown smooth regression parameter
function that varies with the level of quantile. The objective is to test that
the regression parameter is constant across several quantile levels of
interest. The parameter function is estimated by combining ideas from
functional principal component analysis and quantile regression. An adjusted
Wald testing procedure is proposed for this hypothesis of interest, and its
chi-square asymptotic null distribution is derived. The testing procedure is
investigated numerically in simulations involving sparse and noisy functional
covariates and in a capital bike share data application. The proposed approach
is easy to implement and the {\tt R} code is published online at
\url{https://github.com/xylimeng/fQR-testing}.
