my $grammar = <<'END_OF_GRAMMAR';
    {
        use 5.012; #enable say()
        use Data::Dumper;
    }

    startrule: paragraph(s)

    paragraph:  infobox
                { say Dumper(\@item); }
                | word(s)

    infobox:    '{{Infobox' 
                inner_block(s)
                '}}' 
                { 
                    my $inner_blocks = join "", @{$item[2]}; 
                    $return = join "\n", $item[1], 
                                         $inner_blocks,  
                                         $item[4];
                }

    inner_block: brace_block
                 | word(s)
                 { $return = join "\n", @{$item[1]} ; }

    brace_block:    <rulevar: ($lbraces, $rbraces)>
    brace_block:    lbrace(2..) 
                    { 
                        $lbraces = join '', @{$item[1]};
                        $rbraces = "}" x length $lbraces;
                    }

                    inner_block(s) 

                    "$rbraces" 
                    { 
                        #say Dumper(\@item);
                        my $inner_blocks = join "", @{$item[3]};
                        $return = "$lbraces $inner_blocks $rbraces";
                    }

    word:    m{ [^{}]+ }xms

    lbrace: / [{] /xms
                  
END_OF_GRAMMAR

--output:--
$VAR1 = [
          'paragraph',
          '{{Infobox
aaa bbb ccc 
 {{ ddd eee fff
 ggg {{ hhh iii }}jjj }}{{{ kkk {{ lll }}mmm  }}}
'
        ];

$VAR1 = [
          'paragraph',
          '{{Infobox
aaa2 bbb2 ccc2
 {{ ddd2 eee2 fff2 
 ggg2 {{ hhh2 iii2 }}jjj2 }}{{{ kkk2 {{ lll2  }}mmm2  }}}
'
        ];

$VAR1 = [
          'paragraph',
          '{{Infobox
111
'
        ];