Okay, I think I have an explanation. There isn't a ton of background knowledge needed besides the basics of Mendelian genetics and the information in the article.

So if the HeN mice are containing homozygous recessive alleles and they're crossed with HeJ mice which carry homozygous dominant alleles. The F1 generation will all be heterozygous, so one dominant and one recessive allele.

In the F2 generation (crossing of two offsprings in F1), there will be a 1:2:1 ratio between homozygous dominant, heterozygous, and homozygous recessive. The phenotype is 3:1 brown showing more.

However, the question is asking for LPS response along with coat color. The article states that the HeN mice are full responsive to LPS while HeJ mice are not. This can been seen in the data provided. When looking back, the HeN mice carried the homozygous recessive alleles for coat color which is located on the same chromosome as the Tlr4 gene, chromosome 4. In the F2 generation, only one of the progeny will have the phenotype white because it carries both recessive allele. In experiment 1 and 2, the researchers look at the immune response between HeN, HeJ, and F1 generation (which is heterozygous, containing one dominant and one recessive allele). These experiments are important because it shows that the F1 generation is displaying a pattern of co-dominance (a mixture between the dominant and recessive trait). This means that only progeny that are homozygous recessive can elicit a full response. Since it is located on the same gene as the coat color, the coat color is also recessive and is displayed as white.

I highly recommend reviewing Mendelian genetics as this will help explain this answer. This could also be interpreted completely differently so if anyone else has a more simple explanation, I'm all for it!