Appropriate way to visualize significance in 2x3 contingency table using mosaic plot

I've checked multiple threads about handling or visualizing contingency tables, but can't find one that can help my current question. I have a 2x3 contingency table: "group" variable has 3 levels not necessarily ordered, "level" variable has 2 levels (could be 3, but for simplicity, let's say 2). I would like to test independence between group and level variables, and visualize the pairwise relationship between groups as well (group 1 vs 3, 2 vs 3, etc). I'm using vcd::mosaic which runs loglm (which wraps loglin) to visualize individual significance of cells in the contingency table. I like it that the area of each tile reflects the cell frequency and color reflects the statistical significance (residual).

My question is: when I put the full dataset with all 3 groups in loglm, group 3 does not seem to deviate from the null, but when I use the partial datasets with group 1 and 3, or 2 and 3 only, then each pair significantly deviates from the null. What is the best way to perform the test, visualize and interpret the relationship between the 3 groups? Should I use two plots with the pair of groups or one plot with 3 groups? I provide a dummy example dataset and mosaic plot results below:

set.seed(123)
d <- data.frame(group = c(rep(1,20),rep(2,20),rep(3,20)),
level = c(1, rep(0,19),rep(1,19),0, rep(1,10),rep(0,10)))

# 3 groups
tbl123 <- table(d)
vcd::mosaic( ~ group + level, data = tbl123, gp=shading_max, split_vertical=T)

# 2 group
tbl13 <- table(d[d$group %in% c(1,3), ]) vcd::mosaic( ~ group + level, data = tbl13, gp=shading_max, split_vertical=T) tbl23 <- table(d[d$group %in% c(2,3), ])
vcd::mosaic( ~ group + level, data = tbl23, gp=shading_max, split_vertical=T)

Thanks