  We compute the $\bar{h}c$ (pseudo)scalar, (axial-)vector and (axial-)tensor
susceptibilities as a function of $u=m_c/m_h$ between $u=m_c/m_b$ and $u=0.8$
using fully relativistic lattice QCD, employing nonperturbative current
renormalisation and using the second generation 2+1+1 MILC HISQ gluon field
configurations. We include ensembles with $a\approx 0.09\mathrm{fm}$,
$0.06\mathrm{fm}$, $0.045\mathrm{fm}$ and $0.033\mathrm{fm}$ and we are able to
reach the physical $b$-quark on the two finest ensembles. At the physical
$m_h=m_b$ point we find $\overline{m}_b^2 \chi_{1^+}={0.720(34)\times
10^{-2}}$, $\overline{m}_b^2 \chi_{1^-}={1.161(54)\times 10^{-2}}$,
$\chi_{0^-}={2.374(33)\times 10^{-2}}$, $\chi_{0^+}={0.609(14)\times 10^{-2}}$.
Our results for the (pseudo)scalar, vector and axial-vector are compatible with
the expected small size of nonperturbative effects at $u=m_c/m_b$. We also give
the first nonperturbative determination of the tensor susceptibilities, finding
$\overline{m}_b^2 \chi_{T}={0.891(44)\times 10^{-2}}$ and $\overline{m}_b^2
\chi_{AT}={0.441(33)\times 10^{-2}}$. Our value of $\overline{m}_b^2\chi_{AT}$
is in good agreement with the $\mathcal{O}(\alpha_s)$ perturbation theory,
while our result for $\overline{m}_b^2\chi_{T}$ is in tension with the
$\mathcal{O}(\alpha_s)$ perturbation theory at the level of $2\sigma$. These
results will allow for dispersively bounded parameterisations to be employed
using lattice inputs for the full set of $h\to c$ semileptonic form factors in
future calculations, for heavy-quark masses in the range $1.25\times m_c \leq
m_h \leq m_b$.
