I have 2 comments: 


1) Empirically I have a guess for the positive real roots:
Let a scaling-factor $c=\frac1{\ln 2}-1 \approx 0.442$ then
$$ \rho_k \approx (k+3) \cdot c  $$
or $$ k \lt {\rho_k \over c }-3 \lt k+1 \tag {for $k \ge$ 5}$$
I checked this up to $k=5000$ so far using the bernoulli-polynomials and internal float precision of 400 digits in Pari/GP (see table below).

2) looking at the equation modulo $k$ it seems, the lhs of the equation is always equivalent to zero, so solutions can only exist, if also the rhs is zero modulo $k$ - here I assume only odd $k$ so far.    
Let $w$ be the squarefree kernel of primefactors (all are odd) of $k$. Then the rhs is equivalent zero only if $x$ is a multiple of $w$ and thus only such $x$ can solve the equality.      

I hoped to combine the two empirical results somehow, but don't see any further useful property - so also I do not know, whether it is at all worth the effort to actually prove observation 1).      
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Here is a list of the roots, scaled by the scaling factor and the integer value $3$ and the fractional value separated:
  $$\small \begin{array} {r|l}
 k & {\rho_k \over c} &  {\rho_k \over c}=k+{1.5 \over c}+ \epsilon \\
\hline
 4 & 4 + 4.16684831867 & 4 + 1.5/c +        0.344643086811 \\ 
 5 & 5 + 3.65764105650 & 5 + 1.5/c +        0.119219557064 \\ 
 6 & 6 + 3.48606997990 & 6 + 1.5/c +       0.0432658922944 \\ 
 7 & 7 + 3.4241172248  & 7 + 1.5/c +       0.0158397148376 \\ 
 8 & 8 + 3.4015117133  & 8 + 1.5/c +      0.00583236702541 \\ 
 9 & 9 + 3.3933253612  & 9 + 1.5/c +      0.00220830950877 \\ 
 10 & 10+3.3903889002 & 10 + 1.5/c +     0.000908352824271 \\ 
 11 & 11+3.3893375033 & 11 + 1.5/c +     0.000442904593529 \\ 
 12 & 12+3.3889540764 & 12 + 1.5/c +     0.000273163432055 \\ 
 13 & 13+3.3888054311 & 13 + 1.5/c +     0.000207358901590 \\ 
 14 & 14+3.3887395158 & 14 + 1.5/c +     0.000178178523631 \\ 
 15 & 15+3.3887034724 & 15 + 1.5/c +     0.000162222283615 \\ 
 \ldots & \ldots \\
 497 & 497+3.388349245 & 497 + 1.5/c + 0.00000540736229828 \\ 
 498 & 498+3.388349220 & 498 + 1.5/c + 0.00000539654486254\\ 
 499 & 499+3.388349196 & 499 + 1.5/c +  0.00000538577062079\\ 
 500 & 500+3.388349172 & 500 + 1.5/c + 0.00000537503931484\\
 \end{array}
$$