How to create correlated Latin Hypercube samples?

I have been trying to generate 2-d correlated Latin Hypercube samples using a 1-factor Gaussian copula model. However, the resulting correlated quantiles do not necessarily have 1 sample point in each row-column in the 2-d Latin cube.

Below is an example R code using a 2-d case. I've coded a simple LHS random number generator and have not used an R package for that, but I believe this should not matter. The plot shows the 2-d latin cube for uncorrelated (black points) and correlated (red points) samples. You can see that although this seems correct, the correlated samples do not occur at each row/column as it should be for in LHS.

rnlhs <- function(n) {

  # define strata
  brks = seq( 0, 1,length.out = (n+1) )
  strata = data.frame( start = brks[1:n], end = brks[2:(n+1)] )

  s <- runif(n, strata$start, strata$end);

  # reshuffle
  sample(s)

}

rho = 0.5; n = 10

### Random case
set.seed(9)
q1 = rnlhs(n)
q2 = rnlhs(n)

plot(q1,q2, xlim=c(0,1), ylim=c(0,1))
abline(h=seq(0,1,0.1), v=seq(0,1,0.1), col="gray", lty=3)

### Correlated samples (factor model)
q1 = rnlhs(n)
q2 = rnlhs(n)
q0 = rnlhs(n)
X0 = qnorm(q0, 0, 1)
X1 = qnorm(q1, 0, 1)
X2 = qnorm(q2, 0, 1)

Y1 <- sqrt(rho)*X0 + sqrt(1-rho)*X1
Y2 <- sqrt(rho)*X0 + sqrt(1-rho)*X2

# correlated quantiles using the normal CDF
q11 <- pnorm(Y1, 0, 1)
q22 <- pnorm(Y2, 0, 1)

points(q11,q22, col = "red", pch=16)```



The resulting correlated quantiles, q11 and q22, cannot be really defined as Latin Hypercube samples since -for example- both q11, q22 do not have a sample point between [0.1-0.2]. How is it possible to insure that one correlated samples point occurs in each row and column?