Improper syntax highlight for dollar sign $

Describe the bug

The highlight for the dollar sign $ is improper

Steps to reproduce

1. open the following example org file test.org

#+title: Test

example 1: the verbatim should be respected
=$= the text should not be highlighted =$=

example 2: text surrounded by dolloar signs that are distanced by over 2 lines should not be highlighted
$2 billion
hello
oh not good
$3 billion

Expected behavior

1. when $ is closed by two =, i.e =$= then verbatim should be respected. i.e there should be no syntax highlight
2. when two $ are distanced by more two lines, text between them should not be highlighted.

Emacs functionality

see description here
https://orgmode.org/manual/LaTeX-fragments.html

Text within the usual LaTeX math delimiters. To avoid conflicts with currency specifications, single ‘$’ characters are only recognized as math delimiters if the enclosed text contains at most two line breaks, is directly attached to the ‘$’ characters with no whitespace in between, and if the closing ‘$’ is followed by whitespace, punctuation or a dash. For the other delimiters, there is no such restriction, so when in doubt, use ‘(...)’ as inline math delimiters.

Minimal init.lua

vim.cmd [[set runtimepath=$VIMRUNTIME]]
vim.cmd [[set packpath=/tmp/nvim/site]]

local package_root = '/tmp/nvim/site/pack'
local install_path = package_root .. '/packer/start/packer.nvim'

local function load_plugins()
    require('packer').startup {
        {
            'wbthomason/packer.nvim',
            { 'nvim-treesitter/nvim-treesitter' },
            { 'kristijanhusak/orgmode.nvim', branch = 'master' },
        },
        config = {
            package_root = package_root,
            compile_path = install_path .. '/plugin/packer_compiled.lua',
        },
    }
end

_G.load_config = function()
    require('orgmode').setup_ts_grammar()
    require('nvim-treesitter.configs').setup {
        ensure_installed = { 'latex' },
        highlight = {
            enable = true,
            additional_vim_regex_highlighting = { 'org' },
        },
    }

    vim.cmd [[packadd nvim-treesitter]]
    vim.cmd [[runtime plugin/nvim-treesitter.lua]]
    vim.cmd [[TSUpdateSync org]]

    -- Close packer after install
    if vim.bo.filetype == 'packer' then
        vim.api.nvim_win_close(0, true)
    end

    require('orgmode').setup {
        org_highlight_latex_and_related = 'native',
    }

    -- Reload current file if it's org file to reload tree-sitter
    if vim.bo.filetype == 'org' then
        vim.cmd [[edit!]]
    end
end

if vim.fn.isdirectory(install_path) == 0 then
    vim.fn.system { 'git', 'clone', 'https://github.com/wbthomason/packer.nvim', install_path }
    load_plugins()
    require('packer').sync()
    vim.cmd [[autocmd User PackerCompileDone ++once lua load_config()]]
else
    load_plugins()
    load_config()
end

Screenshots and recordings

the first is neovim

the second is emacs

OS / Distro

macOS 12.6

Neovim version/commit

0.8.1

Additional context

No response

Start with the minimal init.lua and the orgmode setup_ts_grammar and org_highlight_latex_and_related entry points, then compare the Neovim highlighting with the stated Emacs LaTeX-fragment rules. Reproduce both examples in test.org. Done means =$= remains verbatim and dollar-delimited text spanning more than two line breaks is not highlighted.