  A $\mu$-constrained Boolean Max-CSP$(\psi)$ instance is a Boolean Max-CSP
instance on predicate $\psi:\{0,1\}^r \to \{0,1\}$ where the objective is to
find a labeling of relative weight exactly $\mu$ that maximizes the fraction of
satisfied constraints. In this work, we study the approximability of
constrained Boolean Max-CSPs via SDP hierarchies by relating the integrality
gap of Max-CSP $(\psi)$ to its $\mu$-dependent approximation curve.
  Formally, assuming the Small-Set Expansion Hypothesis, we show that it is
NP-hard to approximate $\mu$-constrained instances of Max-CSP($\psi$) up to
factor ${\sf Gap}_{\ell,\mu}(\psi)/\log(1/\mu)^2$ (ignoring factors depending
on $r$) for any $\ell \geq \ell(\mu,r)$. Here, ${\sf Gap}_{\ell,\mu}(\psi)$ is
the optimal integrality gap of $\ell$-round Lasserre relaxation for
$\mu$-constrained Max-CSP($\psi$) instances.
  Our results are derived by combining the framework of Raghavendra [STOC 2008]
along with more recent advances in rounding Lasserre relaxations and reductions
from the Small-Set Expansion (SSE) problem. A crucial component of our
reduction is a novel way of composing generic bias-dependent dictatorship tests
with SSE, which could be of independent interest.
