Looking at your css file, you have different font-weight for headers in edit and preview mode.
Change this
.cm-header-1,
.markdown-preview-view h1 {
font-weight: 450;
color: rgb(123, 108, 214);
/* was(115, 98, 205); */
}
into this
.cm-s-obsidian .cm-header-1,
.markdown-preview-view h1 {
font-weight: 450;
color: rgb(123, 108, 214);
/* was(115, 98, 205); */
}
and do the same for all lower headers too (simply add .cm-s-obsidian for each). Currently this header style is overwritten in edit mode by original Obsidian css with font-weight: 600.
And after that remove
.cm-s-obsidian .cm-header {
-webkit-font-smoothing: antialiased;
}