Purrr - conditionally mutate a column in a list of data frames when it exists

Consider the following list of data frames:

library(tidyverse)

df1 <- tibble(
  id = 1:5,
  A = LETTERS[1:5],
  B = letters[10:14]
)
df2 <- tibble(
  id = 1:3,
  A = LETTERS[1:3],
  B = paste(LETTERS[1:3], letters[10:12])
)
df3 <- tibble(
  id = 1:6,
  B = paste(LETTERS[1:6], letters[10:15])
)
df4 <- tibble(
  id = 1:4,
  C = paste(LETTERS[15:18], letters[20:23])
)

df_ls <- list(df1, df2, df3, df4) %>% 
  set_names(paste0("df", 1:4))

I would like to concatenate the elements of A and B into the B column if that's not already the case. Note that not all the data frame have a B column.

The conditions to do this are as follow:

1. the data frame must have both A and B columns
2. the first letter in B must be different than that of A

I'm working with map functions. My attempt so far (without "condition 2"):

df_ls %>% 
  map(
    ~ .x %>% 
      mutate_at(
        vars(matches("B")),
        ~ {
          if (c("A", "B") %in% colnames(.) %>% sum() == 2)
            paste(A, B)
          else
            B
        }
      )
  )

It doesn't work.

Also, I don't manage to write my second condition. I tried & setequal(. %>% pull(A), . %>% pull(B) %>% word(1)), without success.

Edit:
I need to keep all the data frames separately. Only the B column in df1 should be rewritten. df2, df3 and df4 should remain unchanged.
The expected output is:

$df1
# A tibble: 5 x 3
   id A     B
<int> <chr> <chr>
1     1 A     A j
2     2 B     B k
3     3 C     C l
4     4 D     D m
5     5 E     E n   

$df2
# A tibble: 3 x 3
     id A     B    
  <int> <chr> <chr>
1     1 A     A j  
2     2 B     B k  
3     3 C     C l  

$df3
# A tibble: 6 x 2
     id B    
  <int> <chr>
1     1 A j  
2     2 B k  
3     3 C l  
4     4 D m  
5     5 E n  
6     6 F o  

$df4
# A tibble: 4 x 2
     id C    
  <int> <chr>
1     1 O t  
2     2 P u  
3     3 Q v  
4     4 R w