  We consider an $n\times n$ system of ODEs on $\mathbb{P}^1$ with a simple
pole $A$ at $z=0$ and a double pole $u={\rm diag}(u_1, \dots, u_n)$ at
$z=\infty$. This is the simplest situation in which the monodromy data of the
system are described by upper and lower triangular Stokes matrices $S_\pm$, and
we impose reality conditions which imply $S_-=S_+^\dagger$. We study leading
WKB exponents of Stokes matrices in parametrizations given by generalized
minors and by spectral coordinates, and we show that for $u$ on the caterpillar
line (which corresponds to the limit $(u_{j+1}-u_j)/(u_j - u_{j-1}) \to \infty$
for $j=2, \cdots, n-1$), the real parts of these exponents are given by periods
of certain cycles on the degenerate spectral curve $\Gamma(u_{\rm cat}(t), A)$.
  These cycles admit unique deformations for $u$ near the caterpillar line.
Using the spectral network theory, we give for $n=2$, and $n=3$ exact WKB
predictions for asymptotics of generalized minors in terms of periods of these
cycles. Boalch's theorem from Poisson geometry implies that real parts of
leading WKB exponents satisfy the rhombus (or interlacing) inequalities. We
show that these inequalities are in correspondence with finite webs of the
canonical foliation on the root curve $\Gamma^r(u, A)$, and that they follow
from the positivity of the corresponding periods. We conjecture that a similar
mechanism applies for $n>3$.
  We also outline the relation of the spectral coordinates with the cluster
structures considered by Goncharov-Shen, and with ${\mathcal N}=2$
supersymmetric quantum field theories in dimension four associated with some
simple quivers.
