Is where a way to find close pairs within a specific box eround each point in spatstat?

I'm working with spatstat package in R.
I have marked point patterns with mark variables Name, boxA, boxB. BoxA and boxB are dimensions of a rectangle around each point (bounding boxes).
here is an example:

>split(u) # splits pattern in two classes of cells accord. to Name

Point pattern split by factor
 
negat:
Marked planar point pattern: 843917 points
Mark variables: Name, boxA, boxB 
window: rectangle = [3164, 66924] x [25085, 84325] units

posit:
Marked planar point pattern: 93569 points
Mark variables: Name, boxA, boxB 
window: rectangle = [3164, 66924] x [25085, 84325] units

I want to delete points which lay inside the bounding boxes of other points. Previously I did it using closepairs and crosspairs and a fixed radius for neighbouring points like this:

for (i in 1:length(names(u))) {
  if (names(u)[i] == "posit"){ R <- 20 }else{R <- 13} 
     close_list = closepairs(u[[i]], rmax = R, twice = FALSE)
     close_index <- close_list$j
        if(is.empty(close_index) == F){ u[[i]] <- u[[i]][-close_index] } 
                               }
  close_list <- NULL
  close_list = crosspairs(u$negat, u$posit, 13)
  close_index <- close_list$i
  if(is.empty(close_index) == F){u$negat <- u$negat[-close_index]}

I used different radius for both classes of points because their bounding boxes differed in size. However now I have points with very different boxes within each class.
Is where a build-in function in spatstat to determine radius for the "close_list" for each point individually according to boxA or boxB or even to use for each point rectangles with their box dimensions instead of circles?

@Adrian Baddeley offered a solution below to which I added a nested loop to find the numbers of points with overlapping boxes. Here is a small working example in which bounding boxes of points 3 and 4 overlap too much:

# pattern
X<-as.ppp(cbind(c(1,2.5,2,3),c(1,1,3,3)),c(0,5,0,5))

# marks (size of the bounding boxes (BB) around points)
M<-cbind("boxA"=c(1.5,2,4,3),"boxB"=c(1.5,2,4,3))
M<-as.data.frame(M)
X$marks<-M

# find and delete points inside BB of other points
df <- as.data.frame(X)
dx <- with(df, outer(x, x, "-"))
dy <- with(df, outer(y,y,"-"))
conflict <- (abs(dx) < df$boxA/2) & (abs(dy) < df$boxB/2)
diag(conflict) <- FALSE
conflict[upper.tri(conflict)] <- FALSE 
index<-c()
for(i in 1:dim(conflict)[1]){
   for(j in 1:dim(conflict)[2]){
      if(conflict[i,j] == TRUE){
      index <- c(index, i)}
   }
}
X<-X[-index]

This works fine with the small test pattern. I'll test it on real data though.

If all the boxes have the same aspect ratio (ratio of height to width) then you could use nndist.ppp or pairdist.ppp or nncross.ppp using the argument metric to specify a rectangular distance metric (see the help file and examples for convexmetric). For example if the aspect ratio is 1.5 then

m <- convexmetric(owin(c(-1,1)/2, c(-1.5, 1.5)/2)
D <- pairdist(X, metric=m)

gives you the rectangular distances between all pairs of points.

If the boxes can have different aspect ratios, then there is no existing support in spatstat for this calculation. I would recommend converting the point pattern data to a data frame using as.data.frame and then using outer to perform the calculation.