Binomial mid-p value

Question

I've been under the impression that the mid-$p$ values generally control the Type I error, and consequently confidence intervals based on mid-$p$ values control the coverage. However I have checked that this is not true with the binom.midp() function of the binomSamSize package (see below).

Is it a problem with binom.midp()?

library(binomSamSize)

coverage <- function(p, n, conf){
    bounds <- binom.midp(0:n,n,conf)
    low <- bounds$lower
    up <- bounds$upper
    covered <- which(low<p & p <up)-1
    sum(dbinom(covered, n, p)) 
}

n <- 100
conf <- 95/100
p <- seq(0.01,0.99,length.out=100) 

cover <- sapply(p, function(p){
    coverage(p,n,conf)
    }
)

plot(p,cover,type="l")
abline(h=conf, lty=2)

EDIT: Now I have checked the significance level of the hypothesis test based on the mid-p value (very easy to implement), and the results are in agreement:

### mid-p value for H0:{p>p0} 
pv <- function(x,n,p0){
    pbinom(x, n, p0)-dbinom(x,n,p0)/2 
}
### typeI error
typeI <- function(p,n,alpha){
    pvs <- pv(0:n,n,p)
    rejected <- which(pvs<alpha)-1
sum(dbinom(rejected,n,p))
}

### test 
n <- 100
alpha <- 5/100
p <- seq(0.01,0.99,length.out=100) 

probs <- sapply(p, function(p){
    typeI(p,n,alpha)
    }
)

plot(p,probs,type="l")
abline(h=alpha,lty=2)