How to move pixels at the same time in x86 assembly?

Question

I have to make a tron like game with two players where one player moves with wasd keys and the other with arrow keys.

I have the following logic in the keyboard handling

move_players proc
    check_plyr1:
        mov ah, 01h ; check for input
        int 16h
        jz move_end ; if no input is jump to check plyr2

        mov ah,00h
        int 16h ; get key from buffer

        cmp ah, 1eh ; 'a' is pressed
        je move_plyr1_left

        ; and the other keys

        jmp check_plyr2

        move_plyr1_left:
           call plyr1_trace ; places its previous position

            mov ax, [plyr1X]
            dec ax
            mov [plyr1X], ax

            call plyr1_check ;checks collision
            call plyr1_draw ; draws actual position

            jmp check_plyr2

         plyr1_check:
            mov ah,0Dh
            mov cx,[plyr1X] 
            mov dx,[plyr1Y]
            int 10H ; AL = COLOR

            cmp al, 0h
            jne plyr1_lost ; endgame scene

            ret

    check_plyr2:

        cmp ah, 75 ; 'left arrow' is pressed
        je move_plyr2_left
        
        ;and the other keys

        jmp move_end

        move_plyr2_left:
            call plyr2_trace ; places its previous position

            mov ax, [plyr2X]
            dec ax
            mov [plyr2X], ax

            call plyr2_check ;checks collision
            ;call plyr2_draw ; draws actual position

            jmp move_end
      
       plyr2_check:
            mov ah,0Dh
            mov cx,[plyr2X] 
            mov dx,[plyr2Y]
            int 10H ; AL = COLOR

            cmp al, 0h
            jne plyr2_lost ; end scene

            ret

    
    move_end:
        ret
move_players ENDP

the problem is that I press the 'a' button and hold it then it starts to go to the left. Then I press and hold the left arrow without releasing 'a' key. The player1 (controlled by the wasd) stops and only player2 (controlled by arrows move). At this point if I release and press 'a' again then player1 move only again How can I make them move at the same time?

I tried to implement a system where I keep a record of the last key pressed and ignore it if it pressed again but didn't helped only halted the movement of the first player controlled.

I also tried to separate the two inputs in two procedures but that had the same result.