postgis - For Loop Maximum Minimum Distance Between Two Lines

I have a table of line/lane ids in a table called matched_lanes, and I am trying to create a lateral query that on each id in the table to find the maximumum minimum distance between each point.

I.e. for every lane:

Loop through each point on the lane, and find the minimum distance to its first_left_boundary (this is just another lane stored in the matched_lanes table.

Return the MAXIMUM of these minimum distance and append this to our table as max_left_lane

Essentially if you consider these lanes as roads, at any point along the road you could calculate the smallest distance to your (the roads) left neighbour (another road). For each point along the lane this smallest distance can be calculated. I want to find the max of these min distances value for each lane.

I have the below, but I'm not sure it is actually calculating the minimum, but rather the max

Refernces:

How to calculate the average distance among set of points as measure of closeness

Get the lines between all points of a polygon in PostGis : avoid nested loop?

       SELECT 
            lane_id,
            first_left_boundary
            sub.max_lane as max_left_lane
        FROM 
            matched_lanes,

        LATERAL(
            WITH 
                points1 AS(
                    SELECT
                        (ST_DumpPoints(lane_info.geometry)).*  
                    FROM 
                        lane lane_info
                    WHERE 
                        lane_info.id = lane_id
                    ),

                 points2 AS(
                    SELECT 
                        (ST_DumpPoints(first_left_lane.geometry)).*
                    FROM 
                        lane first_left_lane
                    WHERE 
                        first_left_lane.id = first_left_boundary
                )
            SELECT DISTINCT 
                max(ST_DISTANCE(points1.geom, points2.geom))

            FROM 
                points1,
                points2
            )
        AS sub(max_lane)
        ORDER BY 
            lane_id;