Overall Multiple Regression (F test) is not significant but individual regression coefficients are, when controlling for covariates/confounders I am interested in performing a multiple linear regression in order to determine if levels of my protein of interest is associated with cognitive decline in pre-symptomatic Alzheimer's disease. However, based on the literature I have decided to add several covariates/ confounders (such as age, gender, years of education and ApoE4 carrier status) to the model. When I run the analysis, the overall model (F-statistic from ANOVA table) is not significant (p = 0.069). From what I understand, this suggests that none of the regression coefficients differ from 0. However, when I look at the significance of the regression coefficient for my protein of interest, it is highly significant (p = 0.008) when adjusting for the covariates/confounders.
Is it possible that the non-significance of the overall model is due to the the non-significance of the regression coefficients of some of these confounders? For instance, the significance of the  regression coefficients are: ApoE4 (p = 0.855), age (p = 0.180), gender (p = 0.085) education (p = 0.469).
I am only interested in the effects of my protein (p = 0.008) on my dependent variable (cognitive decline) while controlling for these covariates/ confounders.  I am not necessarily interested in the individual regression coefficients of these covariates/ confounders.
Can I still say that there is a significant relationship between my protein and cognitive decline? Multicollinearity and heteroskedasticity do not appear to be present
Thank you

From An Introduction to Statistical Learning, second edition (James et al, 2021):
Given these individual p-values for each variable, why do we need to look
at the overall F-statistic? After all, it seems likely that if any one of the
p-values for the individual variables is very small, then at least one of the
predictors is related to the response. However, this logic is flawed, especially when the number of predictors p is large.
For instance, consider an example in which p = 100 and H0 : β1 = β2 =
··· = βp = 0 is true, so no variable is truly associated with the response. In
this situation, about 5 % of the p-values associated with each variable (of
the type shown in Table 3.4) will be below 0.05 by chance. In other words,
we expect to see approximately five small p-values even in the absence of
any true association between the predictors and the response.8 In fact, it
is likely that we will observe at least one p-value below 0.05 by chance!
Hence, if we use the individual t-statistics and associated p-values in order
to decide whether or not there is any association between the variables and
the response, there is a very high chance that we will incorrectly conclude
that there is a relationship. However, the F-statistic does not suffer from
this problem because it adjusts for the number of predictors. Hence, if H0
is true, there is only a 5 % chance that the F-statistic will result in a pvalue below 0.05, regardless of the number of predictors or the number of
observations.
So, I would say that your model does not provide reliable results about the individual significance of protein. For further details, check chapter 3 of the aforementioned book.
